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Acta Mathematica Academiae Paedagogicae Nyı́regyháziensis 17 (2001), 151–153 www.emis.de/journals SEMISIMPLE COMPLETED GROUP RINGS I. FECHETE Abstract. We give in this note a characterization of semisimple completed group rings. Our result extends theorem of Connell concerning the semisimplicity of group rings to the topological case. 1. Preliminaries By a semisimple ring we mean a ring semisimple in the sense of Jacobson. By a profinite ring (group) we mean a compact totally disconnected ring (group). Let R be a profinite ring with identity and G a profinite group. If V is a two sided ideal of R and N an invariant subgroup of G, consider the two sided ideal of the group ring R [G] , (V, N ) = V [G] + (1 − N ) , where (1 − N ) is the ideal of R [G] generated by the set 1 − N and ( n ) X ∗ V [G] = vi gi : n ∈ N , v1 , . . . , vk ∈ V, g1 , . . . , gk ∈ G . i=1 We will remind for convenience some results from [5]. Lemma 1.1. Let R, R0 are two rings with identity and G, G0 two groups. If f : R → R0 is a ring homomorphism and ϕ : G → G0 is a group homomorphism, then the kernel of the canonical homomorphism λ : R [G] → R0 [G0 ] extending f and ϕ is (V, N ), where V = kerf and N = kerϕ. Lemma 1.2. Let R be a profinite ring with identity and G a profinite group. Then ∩ (V, N ) = {0} , where V runs all open ideals of R and N all open invariant subgroups of G. A Hausdorff topological ring (R, T) is said to be totally bounded provided for every neighborhood V of zero there exists a finite subset F such that R = F + V . It is well known that a Hausdorff topological ring is totally bounded if and only if b is compact. b T its completition R, For a profinite ring R with identity and a profinite group G, the family {(V, N )} where V runs all open ideals of R and N all open invariant subgroups of G gives a totally bounded ring topology T on R [G]. The completion of the topological ring (R [G] , T) is called the completed group ring (and is denoted by R [[G]] ). Lemma 1.3. For each closed invariant subgroup N of G the ideal (1 − N ) of R [G] is closed. 2000 Mathematics Subject Classification. 16W80. Key words and phrases. Semisimple ring, completed group ring. 151 152 I. FECHETE Lemma 1.4. For each closed invariant subgroup N of G holds . R [[G]] (1 − N ) ∼ = R [[G/N ]] Theorem 1.5. Let R be a profinite ring with identity and G a profinite group. If {Nα }α∈Ω is a filter base consisting of closed invariant subgroups of G, then R [[G]] is the inverse limit of rings R [[G/Nα ]] and the canonical projections are onto. Remark 1.1. Theorem 1.5 and lemma 1.4 show that our definition of completed group ring is equivalent with the definition given in [3], through the inverse limit of the rings R [G/N ] , where N runs all open invariant subgroups of G. Question. For which profinite rings R and profinite groups G is the topological ring (R [G] , T) minimal? Theorem 1.6 (The universal property of completed group rings). Let Λ be a profinite commutative ring with identity and G a profinite group. If A is a compact Λ-algebra with identity, then each continuous homomorphism α of G in U (A) can be extended to a continuous homomorphism α b of Λ [[G]] in A. 0 Remark 1.2. If α : G → G is a continuous homomorphism of a profinite group G in a profinite group G0 and R is a profinite ring then there exists a continuous homomorphism α b : R [[G]] → R [[G0 ]] extending α. The following result of Connell [1] yields a characterization of semisimple group rings: Theorem 1.7. The group ring R [G] is semisimple if and only if the following conditions are satisfied: (1) R is a semisimple ring; (2) G is a finite group; (3) the order of G is a unit in R. 2. The main result Theorem 2.1. Let R be a profinite ring with identity and G a profinite group. The completed group ring R [[G]] is semisimple if and only if are satisfied the following two conditions: (1) R is semisimple; (2) for every open ideal V of R and for every open invariant subgroup N of G the order of G/N is a unit in R/V . Proof. Suppose that R [[G]] is semisimple. By theorem of Kaplansky [2], Y R [[G]] ∼ (Fi )ni , = i∈I where each Fi is a finite field, ni a natural number and (Fi )ni is the ring of ni × ni matrices over Fi . In particular, R [[G]] is a regular ring in the sense of von Neumann. By remark 1.2 R is a continuous homomorphic image of R [[G]] , therefore R is regular, hence semisimple. If N is an open invariant subgroup of G and V an open ideal of R, then by lemma 1.1 there is a homomorphism α of R [G] onto (R/V ) [G/N ]. Since ker α = (V, N ) , α is continuous. We extend by the continuity the homomorphism α to a homomorphism α b of R [[G]] onto (R/V ) [G/N ]. Since R [[G]] is regular, the ring (R/V ) [G/N ] is regular too, therefore it is semisimple. According to theorem of Connell, the order of G/N is a unit in R/V . Conversely, assume that R and G satisfy the conditions of theorem. We claim that the group ring R [G/N ] is semisimple for each open invariant subgroup N of G. For each open ideal V of R, fV denotes the canonical homomorphism of R [G/N ] in (R/V ) [G/N ]. By the condition of theorem, (R/V ) [G/N ] is semisimple. Since SEMISIMPLE COMPLETED GROUP RINGS 153 ∩IV = {0}, where IV = ker fV , the ring R [G/N ] is semisimple as a subdirect product of semisimple rings. For each open invariant subgroup N of G, λN denotes the canonical homomorphism R [[G]] → R [G/N ]. The completed group ring R [[G]] is according to lemma 1.4 a subdirect product of rings R [G/N ] where N runs all open invariant subgroups of G. We get that R [[G]] is semisimple. Corollary 2.2. If K is a finite field, then the completed group ring K [[G]] is semisimple if and only if for every open invariant subgroup N of G, char (K) |G/N |. Acknowledgments. I wish to express my gratitude to Prof. M.I. Ursul for his interest to results of this paper. References [1] I.G. Connell. On the group ring. Can. J. Math., 15:650–685, 1963. (See J. Lambek, Lectures on Rings and Modules, Blaisdell Publishing, 1966). [2] I. Kaplansky. Topological rings. American Journal of Mathematics, 69:153–183, 1947. [3] H. Koch. Galoissche Theorie der p-Erweiterungen, volume 10 of Math. Monogr. VEB, Berlin, 1970. [4] M. Ursul. Topological groups and rings. Ed. Univ. Oradea, 1998. [5] M. Ursul and I. Fechete. Completed commutative group rings without zero divisors. Bul. A. S. a R. M., 2001. (to appear). Received January 8, 2001. Department of Mathematics University of Oradea Armatei Romane, 5 3700, Oradea, Romania E-mail address: [email protected]